Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

A Novel Radially Closable Tubular Origami Structure (RC-ori) for Valves

Cylindrical Kresling origami structures are often used in engineering fields due to their axial stretchability, tunable stiffness, and bistability, while their radial closability is rarely mentioned to date. This feature enables a valvelike function, which inspired this study to develop a new origami-based valve. With the unique one-piece structure of origami, the valve requires fewer parts, which can improve its tightness and reduce the cleaning process. These advantages meet the requirements of sanitary valves used in industries such as the pharmaceutical industry. This paper summarizes the geometric definition of the Kresling pattern as developed in previous studies and reveals the similarity of its twisting motion to the widely utilized iris valves. Through this analogy, the Kresling structure’s closability and geometric conditions are characterized. To facilitate the operation of the valve, we optimize the existing structure and create a new crease pattern, RC-ori. This novel design enables an entirely closed state without twisting. In addition, a simplified modeling method is proposed in this paper for the non-rigid foldable cylindrical origami. The relationship between the open area and the unfolded length of the RC-ori structure is explored based on the modeling method with a comparison with nonlinear FEA simulations. Not only limited to valves, the new crease pattern could also be applied to microreactors, drug carriers, samplers, and foldable furniture